#6346734

order	512
nilpotency class	3
isoclinism	isoclinic class of #6346727
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#62820	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

3072 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2,6,4)(3,11,15,27)(5,18,20,30)(7,9,22,16)(8,23,12,14)(10,25,17,21)(13,26,29,31)(19,24,32,28);(1,3)(2,8)(4,12)(5,19)(6,15)(7,13)(9,24)(10,26)(11,18)(14,21)(16,28)(17,31)(20,32)(22,29)(23,25)(27,30);(1,4)(2,6)(3,12)(5,10)(7,16)(8,15)(9,22)(11,14)(13,28)(17,20)(18,21)(19,26)(23,27)(24,29)(25,30)(31,32);(2,9)(4,16)(8,24)(10,18)(11,26)(12,28)(17,30)(27,31);(3,13)(8,24)(11,26)(12,28)(14,19)(15,29)(23,32)(27,31)

Polycyclic presentation:

2^9|a^2=i,f^2=h,b^a=bf,c^a=cg,d^a=dh,f^a=fh,e^b=eh,f^b=fh,g^b=gi,d^c=dh,f^c=fi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6253930);